Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

Abstract

The purpose of this paper is to describe a general procedure for computing analogues of Young’s seminormal representations of the symmetric groups . The method is to generalize the Jucys – Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori – Hecke algebras . The combinatorics of these elements allow one to compute irreducible representations explicitly and often very easily . In this paper we do these computations for Weyl groups and Iwahori – Hecke algebras of types A n , B n , D n , G 2 . Although these computations are within reach for types F 4 , E 6 , and E 7 , we shall , in view of the length of the current paper , postpone this to another work . In reading this paper , I would suggest that the reader begin with § 3 , the symmetric group case , and go back and pick up the generalities from §§ 1 and 2 as they are needed . This will make the motivation for the material in the earlier sections much more clear and the further examples in the later sections very easy . The realization that the Jucys – Murphy elements for Weyl groups and Iwahori – Hecke algebras come from the very natural central elements in (2 . 1) and Proposition 2 . 4 is one of the main points of this paper . There is a simple concrete connection (Proposition 2 . 8) between Jucys – Murphy type elements in Iwahori – Hecke algebras and Jucys – Murphy elements in group algebras of Weyl groups . I know that the analogues of the Jucys – Murphy elements in Weyl groups of types B and D will be new to some of the experts and known to others . These Jucys – Murphy elements for types B and D are not new ; similar elements appear in the paper of Cherednik [ 7 ] , but I was not able to recognize them there until they were pointed out to me by M . Nazarov . I extend my thanks to him for this . Some people were asking me for Jucys – Murphy elements in type G 2 as late as June 1995 . In July 1995 I was told that it was not known how to quantize the elements of Cherednik , that is , to find analogues of them in the Iwahori – Hecke algebras of types B and D . Of course , this had been done already in 1974 , by Hoefsmit . I have chosen to state my results in terms of the general mechanism of path algebras which I have defined in § 1 . This is a technique which I learned from H . Wenzl during our work on the paper [ 30 ] . It is a well-known method in several fields (with many dif ferent terminologies) . I shall mention here only a few of the